Embedded newform 336.3.bv.a.325.46

• Label: 336.3.bv.a.325.46
• Level: $336$
• Weight: $3$
• Character: 336.325
• Analytic conductor: $9.155$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	336.bv (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.15533688251\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			325.46
Character	\(\chi\)	\(=\)	336.325
Dual form			336.3.bv.a.61.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20011 + 1.59991i) q^{2} +(-1.67303 + 0.448288i) q^{3} +(-1.11945 + 3.84016i) q^{4} +(-7.56863 - 2.02801i) q^{5} +(-2.72505 - 2.13871i) q^{6} +(4.78196 - 5.11203i) q^{7} +(-7.48739 + 2.81761i) q^{8} +(2.59808 - 1.50000i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 12 q^{4} - 24 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\)	\(85\)	\(113\)	\(127\)	\(241\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(1\)	\(1\)	\(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20011	+	1.59991i	0.600057	+	0.799957i
							\(3\)	−1.67303	+	0.448288i	−0.557678	+	0.149429i
\(5\)	−7.56863	−	2.02801i	−1.51373	−	0.405601i								−0.596055	−	0.802944i	\(-0.703266\pi\)
							−0.917670	+	0.397342i	\(0.869933\pi\)
\(7\)	4.78196	−	5.11203i	0.683137	−	0.730290i
							\(11\)	17.6924	−	4.74066i	1.60840	−	0.430969i	0.660833	−	0.750533i	\(-0.270203\pi\)
0.947565	+	0.319564i	\(0.103537\pi\)
\(13\)	−11.4249	−	11.4249i	−0.878837	−	0.878837i	0.114578	−	0.993414i	\(-0.463449\pi\)
							−0.993414	+	0.114578i	\(0.963449\pi\)
\(17\)	9.25693	+	5.34449i	0.544526	+	0.314382i	0.746911	−	0.664924i	\(-0.231536\pi\)
							−0.202386	+	0.979306i	\(0.564869\pi\)
\(19\)	−3.45503	+	12.8943i	−0.181844	+	0.678650i	0.813441	+	0.581648i	\(0.197592\pi\)
							−0.995284	+	0.0970016i	\(0.969075\pi\)
\(23\)	21.6976	−	12.5271i	0.943374	−	0.544657i	0.0523573	−	0.998628i	\(-0.483327\pi\)
							0.891016	+	0.453971i	\(0.149993\pi\)
\(29\)	28.0702	−	28.0702i	0.967938	−	0.967938i	−0.0315636	−	0.999502i	\(-0.510049\pi\)
							0.999502	+	0.0315636i	\(0.0100487\pi\)
\(31\)	−34.6012	−	19.9770i	−1.11617	−	0.644421i	−0.175749	−	0.984435i	\(-0.556235\pi\)
							−0.940420	+	0.340014i	\(0.889568\pi\)
\(37\)	7.41047	−	27.6563i	0.200283	−	0.747466i	−0.790553	−	0.612394i	\(-0.790207\pi\)
							0.990836	−	0.135072i	\(-0.0431267\pi\)
\(41\)	44.9167			1.09553			0.547764	−	0.836633i	\(-0.315479\pi\)
							0.547764	+	0.836633i	\(0.315479\pi\)
\(43\)	−51.2964	−	51.2964i	−1.19294	−	1.19294i	−0.976238	−	0.216703i	\(-0.930470\pi\)
							−0.216703	−	0.976238i	\(-0.569530\pi\)
\(47\)	21.4132	−	12.3629i	0.455600	−	0.263041i	−0.254592	−	0.967048i	\(-0.581941\pi\)
							0.710192	+	0.704008i	\(0.248608\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.bv.a.325.46	yes	256
7.5	odd	6	inner	336.3.bv.a.229.40	yes	256
16.13	even	4	inner	336.3.bv.a.157.40	yes	256
112.61	odd	12	inner	336.3.bv.a.61.46	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.bv.a.61.46	✓	256	112.61	odd	12	inner
336.3.bv.a.157.40	yes	256	16.13	even	4	inner
336.3.bv.a.229.40	yes	256	7.5	odd	6	inner
336.3.bv.a.325.46	yes	256	1.1	even	1	trivial